Diffusion

Describing diffusion quantitatively

Exercise: Diffusion

Suppose we have molecule � diffusing in a domain between boundary 1 and boundary 2, where ��,1 and ��,2 are held fixed.

What happens to �� between boundary 1 and boundary 2 and what is the rate of diffusional mass transfer (moles/time) across some area �? Note that � represents an area � by � into the figure.

Here is a particle simulation.

Fick’s law of diffusion

�˙�,�=−���������

For a system in which there is pure diffusion only (no convection, no reactions, constant properties), the concentration profile is linear.

Thus, Fick’s law can be approximated by

�˙�,�=−����(��,2−��,1�2−�1)=−����(Δ��Δ�)

Hence, the driving potential for diffusional mass transfer is ��,2−��,1.

In the above equations,

Variable	Definition	Typical units
�˙�,�	moles of species � transferred per unit time from location 1 to location 2	mol/s
���	the ‘binary diffusivity’ (or diffusion coefficient) of species � through medium �	m2/s
�	the ‘face’ area through which transfer occurs	m2
��,2−��,1	the difference in concentration between locations 2 and 1	mol/m3
�2−�1	the distance between locations 2 and 1	m

As a reminder, the area for mass transfer is not the ‘edge-view’ area, but the ‘face-view’ area:

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Adjustments to the area for mass transfer

When we consider the area for mass transfer, �, we need to consider that some of the apparent area may not be available to transport mass.

For instance, imagine a porous membrane with a pore fraction of �pore. In this case, the actual area available for mass transfer is only the porous fraction of the total, and thus,

�=�apparent�pore

We will generally write � in our governing rate equations, but keep in mind this distinction and include surface area adjustments when necessary.